| L(s) = 1 | − 1.23·2-s − 0.481·4-s + 1.54·5-s − 3.80·7-s + 3.05·8-s − 1.90·10-s + 3.76·11-s − 2.63·13-s + 4.68·14-s − 2.80·16-s − 3.77·17-s + 6.09·19-s − 0.744·20-s − 4.63·22-s − 0.909·23-s − 2.61·25-s + 3.24·26-s + 1.83·28-s + 6.80·29-s − 2.66·32-s + 4.65·34-s − 5.87·35-s − 1.81·37-s − 7.51·38-s + 4.72·40-s − 0.337·41-s + 3.88·43-s + ⋯ |
| L(s) = 1 | − 0.871·2-s − 0.240·4-s + 0.691·5-s − 1.43·7-s + 1.08·8-s − 0.602·10-s + 1.13·11-s − 0.730·13-s + 1.25·14-s − 0.701·16-s − 0.915·17-s + 1.39·19-s − 0.166·20-s − 0.988·22-s − 0.189·23-s − 0.522·25-s + 0.636·26-s + 0.346·28-s + 1.26·29-s − 0.470·32-s + 0.797·34-s − 0.993·35-s − 0.298·37-s − 1.21·38-s + 0.747·40-s − 0.0526·41-s + 0.592·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8865140186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8865140186\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 1.23T + 2T^{2} \) |
| 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 + 3.80T + 7T^{2} \) |
| 11 | \( 1 - 3.76T + 11T^{2} \) |
| 13 | \( 1 + 2.63T + 13T^{2} \) |
| 17 | \( 1 + 3.77T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 + 0.909T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 + 0.337T + 41T^{2} \) |
| 43 | \( 1 - 3.88T + 43T^{2} \) |
| 47 | \( 1 - 1.18T + 47T^{2} \) |
| 53 | \( 1 - 2.34T + 53T^{2} \) |
| 59 | \( 1 + 7.77T + 59T^{2} \) |
| 61 | \( 1 + 2.72T + 61T^{2} \) |
| 67 | \( 1 + 7.42T + 67T^{2} \) |
| 71 | \( 1 - 5.09T + 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 + 9.73T + 79T^{2} \) |
| 83 | \( 1 - 8.39T + 83T^{2} \) |
| 89 | \( 1 + 5.09T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76102595848559008542190342946, −7.11062479679026086816998470195, −6.51320188548594663624822262760, −5.91422868941683068232596470587, −4.98996217672024392502305748755, −4.22866319465634262408542711787, −3.41206003910733068899875080636, −2.52448557110213957051061563805, −1.52878373452986343005696448779, −0.54685361318613941626128335031,
0.54685361318613941626128335031, 1.52878373452986343005696448779, 2.52448557110213957051061563805, 3.41206003910733068899875080636, 4.22866319465634262408542711787, 4.98996217672024392502305748755, 5.91422868941683068232596470587, 6.51320188548594663624822262760, 7.11062479679026086816998470195, 7.76102595848559008542190342946
